A universal tripartite entanglement signature of ungappable edge states

TL;DR

This paper introduces a universal entanglement measure called the Markov gap as a diagnostic tool for ungappable edge states in 2D topological phases, linking it to boundary conformal field theories.

Contribution

It demonstrates that the Markov gap universally detects ungappable edge states and relates it to the boundary's central charge, providing a new bulk entanglement criterion.

Findings

Markov gap equals (c_+/3) log 2 for ungappable edge states.

Markov gap is zero for string-net models.

Numerical verification for Chern insulators shows Markov gap proportional to boundary central charge.

Abstract

Gapped two-dimensional topological phases can feature ungappable edge states which are robust even in the absence of protecting symmetries. In this work we show that a multipartite entanglement measure recently proposed in the context of holography, the Markov gap, provides a universal diagnostic of ungappable edge states. Defined as a difference of the reflected entropy and mutual information $h(A:B) = S_R(A:B) - I(A:B)$ between two parties, we argue that for $A,B$ being adjacent subregions in the bulk, $h=\frac{c_+}{3}\log 2$, where $c_+$ is the minimal total central charge of the boundary theory. As evidence, we prove that $h=0$ for string-net models, and numerically verify that $h=\frac{|C|}{3}\log 2$ for a Chern-$C$ insulator. Our work establishes a unique bulk entanglement criteria for the presence of a conformal field theory on the boundary.